It's pretty clear within this dominion community that reshuffling can have a huge effect on the game. Using council room, I was able to observe the effect reshuffling can have on a game. Specifically in regard to familiar because to purchase familiar, you must have a potion in your hand. Below is a graph of win rate vs. turn card was gained (sorry I don't know how to integrate this into the post, you'll just have to look at the link). At the beginning stages of the game, people will most likely have 1 potion. This means that their potion was in their hand on the turn they gained a familiar.

From this graph, familiar is obviously a very powerful card. But upon closer analysis, you can see dips in the win rate corresponding to turns 5, 7, and 9. Coincidentally these turn numbers (especially 5 and 7) happen in which a reshuffle was necessary to complete the 5-card hand. A familiar gained on those turns indicates that both the potion and the familiar purchased missed a reshuffle. I looked at a few different cards, but familiar was the only one showing that trend. I would guess that since most other cards don't depend on a single card (potion) to purchase they would not be that susceptible to bad shuffling (because you could purchase them anytime you had enough money).

Check the council room section for an idea related to the shuffling mechanic.